#7575568

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,33)(25,36)(30,39)(31,40)(32,44)(35,41)(37,43)(38,42)(45,48)(46,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,35)(21,32)(22,33)(23,31)(25,41)(26,29)(27,40)(28,44)(30,36)(34,39)(37,45)(38,46)(42,48)(43,47);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,38)(22,36)(23,37)(24,39)(26,34)(27,43)(28,42)(29,41)(31,46)(32,45)(33,35)(40,48)(44,47);(10,13)(14,16)(18,26)(19,31)(20,36)(21,23)(24,40)(29,44)(30,45)(32,33)(35,46)(39,47)(41,48)(42,43);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,26,28)(19,32,33,31)(20,37,36,38)(24,40,29,44)(25,42,34,43)(30,46,35,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7482764);